For positive integers $a$, $b$, and $c$, what is the value of the product $abc$? \[ \dfrac {1}{a + \dfrac {1}{b + \dfrac {1}{c}}} = \dfrac38 \]

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julianas4159 julianas4159

07-07-2018
Mathematics

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For positive integers $a$, $b$, and $c$, what is the value of the product $abc$? \[ \dfrac {1}{a + \dfrac {1}{b + \dfrac {1}{c}}} = \dfrac38 \]

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DeanR DeanR · Answer 1 · 2018-07-07T10:08:43+02:00

I don't think the $ signs work as math delimiters. I would be nice if they did.

That looks like a simple continued fraction ("simple" is a technical term meaning the numerators are all 1).

[tex]\dfrac{1}{a + \dfrac {1}{b + \dfrac {1}{c}}} = \dfrac 3 8[/tex]

I could go on for hours about continued fractions. The way we expand a regular fraction as a continued fraction is essentially Euclid's algorithm for the GCD:

[tex]\dfrac{3}{8} = \dfrac{1}{\frac 8 3} = \dfrac{1}{2 + \dfrac 2 3} = \dfrac{1}{2 + \dfrac{1}{\frac 3 2}} = \dfrac{1}{2 + \dfrac{1}{1 + \dfrac 1 2}}[/tex]

So we have a=2, b=1, c=2, a product

abc=4

aartipothineni aartipothineni · Answer 2 · 2020-01-24T14:19:56+01:00

aartipothineni aartipothineni

24-01-2020

Answer:

4

Step-by-step explanation:

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